There are two paradigms of reasoning from evidence at work in the biomedical and social sciences: the experimental and the inferential. The experimental paradigm is currently dominant in all the domains labelled ‘evidence-based’, which include parts of medicine, dentistry, nursing, psychology, education, social policy and criminal justice, but also in parts of development economics. Whereas the experimental paradigm has received considerable philosophical analysis and support since the times of Bacon and Mill (and continues to enjoy attention and support in very recent work on causation and evidence), the inferential paradigm has neither been articulated nor defended. The overall aim of this paper is to fill this gap and develop a ‘theory of inferential judgement’ as an alternative to both the experimental paradigm in the sciences as well as traditional philosophical theories of evidence.